1st Edition

Fractals and Chaos An illustrated course

By Paul S. Addison Copyright 1997
    268 Pages
    by CRC Press

    268 Pages
    by CRC Press

    Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.

    INTRODUCTION
    Introduction
    A matter of fractals
    Deterministic chaos
    Chapter summary and further reading

    REGULAR FRACTALS AND SELF-SIMILARITY
    Introduction
    The Cantor set
    Non-fractal dimensions: the Euclidean and topological dimension
    The similarity dimension
    The Koch curve
    The quadratic Koch curve
    The Koch island
    Curves in the plane with similarity dimension exceeding 2
    The Sierpinski gasket and carpet
    The Menger Sponge
    Chapter summary and further reading
    Revision questions and further tasks

    RANDOM FRACTALS
    Introduction
    Randomizing the Cantor set and Koch curve
    Fractal boundaries
    The box counting dimension and the Hausdorff dimension
    The structured walk technique and the divider dimension
    The perimeter-area relationship
    Chapter summary and further reading
    Revision questions and further tasks

    REGULAR AND FRACTIONAL BROWNIAN MOTION
    Introduction
    Regular Brownian motion
    Fractional Brownian motion: time traces
    Fractional Brownian surfaces
    Fractional Brownian motion: spatial trajectories
    Diffusion limited aggregation
    The color and power and noise
    Chapter summary and further reading
    Revision questions and further tasks

    ITERATIVE FEEDBACK PROCESSES AND CHAOS
    Introduction
    Population growth and the Verhulst model
    The logistic map
    The effect of variation in the control parameter
    General form of the iterated solutions of the logistic map
    Graphical iteration of the logistic map
    Bifurcation, stability and the Feigenbaum number
    A two dimensional map: the Henon model
    Iterations in the complex plane: Julia sets and the Mandelbrot set
    Chapter summary and further reading
    Revision questions and further tasks

    CHAOTIC OSCILLATIONS
    Introduction
    A simple nonlinear mechanical oscillator: the Duffing oscillator
    Chaos in the weather: the Lorenz model
    The Rossler systems
    Phase space, dimension and attractor form
    Spatially extended systems: coupled oscillators
    Spatially extended systems: fluids
    Mathematical routes to chaos and turbulence
    Chapter summary and further reading
    Revision questions and further tasks

    CHARACTERIZING CHAOS
    Introduction
    Preliminary characterization: visual inspection
    Preliminary characterization: frequency spectra
    Characterizing chaos: Lyapunov exponents
    Characterizing chaos: dimension estimates
    Attractor reconstruction
    The embedding dimension for attractor reconstruction
    The effect of noise
    Regions of behavior on the attractor and characterization limitations
    Chapter summary and further reading
    Revision questions and further task

    APPENDIX 1: Computer Program for Lorenz Equations
    APPENDIX 2: Illustrative Papers
    APPENDIX 3: Experimental Chaos

    SOLUTIONS
    REFERENCES

    Biography

    Paul S. Addison

    "Fractals and Chaos: An Illustrated Course is well designed for self-study, making it a great practical resource for those working in the physical sciences or engineering as well as for students."
    —Danny Yee’s Book Reviews, February 2016